6.3 Comparative Results

[1] In the pair of triads shown in Figure 6.3a, one is sampled
from a chorale harmonization (see
§4.1) by J.S. Bach (BWV 253, "Ach bleib' bei uns, Herr Jesu Christ"). The
other triad is randomly generated without regard to rules for doubling or
spacing. (See §4.2.)

Figure 6.3a. One triad sampled from BWV 253, and a random twin.
In
terms of the analysis, it is unknown a priori which is composed.

[2] A priori, we don't know which triad is which. So each triad has an
equal probability of being composed, or of being random. Our statistical models
refine these probabilities by using features of the doubled tones, along with
chord spacing. (See §5.2.)

One version of the model uses the triad member of the doubled tone. (See
§6.1.) In the example above, the left triad doubles the root, and the
right triad doubles the third. Since the root is a bit more likely to be
doubled in a first-inversion major triad, this difference increases the
probability that the left triad is composed. In addition, the left triad has
its largest space between bass and tenor, while the right does not.
Considering both doubling and spacing, this model assigns the left triad a
higher probability of being composed (75%).

Another version of the model uses the scale degree of the doubled tone. (See
§6.2.) In the example above, the left triad doubles the fifth degree, and
the right doubles the leading tone. Combining this with the left triad's
better spacing, this model assigns the left triad a much higher probability of
being composed (92%).

A final version of the model uses both the scale degree and the triad
member of the doubled tone. Combining this information with the left triad's
better spacing, this model assigns the left triad a higher probability of
being composed. The estimated probability (84%) is about midway between the
probabilities assigned by the previous models, which use triad members or
scale degrees alone.(84)

[3] In this case, all three models are accurate. The left triad is in fact
composed. In other cases, however, one model or the other may be inaccurate,
assigning the random triad a higher probability of being composed.

[4] Overall, all three models are about equally accurate (see Figure 6.3b),
identifying the composed triad in about 70% of all pairs.

Figure 6.3b. The accuracy of four models is shown: the spacing rule
alone, the triad-member model,
the scale-degree model, and both models together. All three doubling
models include the chord-spacing
predictor. Results are broken down by genre: quartets (squares) and
chorales (diamonds).

[4] Since the model based on scale degrees is about as accurate as the model
based on triad members, and since there is little improvement from using scale
degrees and triad members together, it seems that the scale-degree and
triad-member doubling rules are redundant with one another. They are merely
different ways of representing the same musical practices.

[5] As remarked in section 1, some redundancies are obvious. In major
dominant chords, for instance, doubling the leading tone is indistinguishable
from doubling the third. Other redundancies are subtler, but probably inevitable
when a large number of features are used to describe a fairly basic musical
practice.